IIRC, for dogs, at least, the #1 cause of food allergies is (was?) chicken, not corn, although corn is also in the list of foods that can give them food allergies.
Let’s start with the duck and potato diet*. Was the dog still getting preventive medicine (flea, tick, and heartworm meds) during the time you tried it? If so, THOSE could’ve been the ones giving the dog allergies. A dog in a very strict food trial is not supposed to eat anything else. This means no rawhides, no outside food sources, no table scraps, no treats, nothing chicken or meat or whatever flavored… and no chewable preventatives.
Also, a dog with allergies really needs to be on flea/tick control, as fleas are one of the main causes of allergies. Have you tried different products, perhaps there is resistance? And by those, I mean veterinarian-prescribed, no OTC ones?
Have they done skin scraping and culture, to rule out fungal, bacterial, and parasitic (mites) infections? If not, they should. They may not have been the original cause of the itching, but if it has been going on for a long time, by now the skin may have several secondary problems, like bacterial, fungal, or parasitic infections. Those would have to be treated then too, despite being secondary. Otherwise the issue may not resolve.
Considering all that you’ve done, I’ll recommend your vet sends you to a dermatologist, or at least a vet who really really loves to deal with skin diseases. And be prepared to answer lots of questions about history and such…
*What kibbles need to bind is a source of starch. If the diet you’re mentioning is the one I remember seeing, prescribed for allergic dogs, then no, those diets are NOT supposed to have regular ingredients (like chicken, or corn). The source of carbs (starch) would be the potatoes, in your case.